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Exchange interaction is the main physical effect responsible for ferromagnetism, and has no classical analogue. For bosons, the exchange symmetry makes them bunch together, and the exchange interaction takes the form of an effective attraction that causes identical particles to be found closer together, as in Bose–Einstein condensation.
This symmetry may be described as the symmetry under the exchange of labels attached to the particles (i.e., to the single-particle Hilbert spaces). Clearly, = (the identity operator), so the eigenvalues of P are +1 and −1. The corresponding eigenvectors are the symmetric and antisymmetric states:
In quantum electrodynamics, the local symmetry group is U(1) and is abelian. In quantum chromodynamics, the local symmetry group is SU(3) and is non-abelian. The electromagnetic interaction is mediated by photons, which have no electric charge. The electromagnetic tensor has an electromagnetic four-potential field possessing gauge symmetry.
The exchange operator acts by switching the labels on any two identical particles described by the joint position quantum state |, . [2] Since the particles are identical, the notion of exchange symmetry requires that the exchange operator be unitary .
Any number of identical bosons can occupy the same quantum state, such as photons produced by a laser, or atoms found in a Bose–Einstein condensate. A more rigorous statement is: under the exchange of two identical particles, the total (many-particle) wave function is antisymmetric for fermions and symmetric for
(It is a non-trivial result of quantum field theory [2] that the exchange of even-spin bosons like the pion (spin 0, Yukawa force) or the graviton (spin 2, gravity) results in forces always attractive, while odd-spin bosons like the gluons (spin 1, strong interaction), the photon (spin 1, electromagnetic force) or the rho meson (spin 1, Yukawa ...
The spin–statistics theorem proves that the observed relationship between the intrinsic spin of a particle (angular momentum not due to the orbital motion) and the quantum particle statistics of collections of such particles is a consequence of the mathematics of quantum mechanics.
The Holstein–Herring formula had limited applications until around 1990 when Kwong-Tin Tang, Jan Peter Toennies, and C. L. Yiu [8] demonstrated that can be a polarized wave function, i.e. an atomic wave function localized at a particular nucleus but perturbed by the other nuclear center, and consequently without apparent gerade or ungerade symmetry, and nonetheless the Holstein–Herring ...